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Stream: learning: questions

Topic: κ-filtered characterization

Robin Jourde (Oct 03 2025 at 16:05):

Dear all,
I am reading Makkai and Paré's Accessible categories and in the first chapter, they define $\kappa$-filtered categories and write the following remark:

Note the easily seen fact that, for $I$ to be $\kappa$-filtered, it suffices to have the condition in 1.1.1 for the following specific graphs $G$: the discrete graph on $\lambda$ elements, and the graph with two nodes and $\lambda$ many parallel arrows, both for $\lambda$ any cardinal $< \kappa$.

where 1.1.1 is "every diagram of source $G$ has a cocone".
I thought I had a proof but as I am looking back on this part, I can't recover it. Do you know how I can prove this ?
in the finite case I see how to do it but for the infinite case it seems impossible to me
Thanks !

Mike Shulman (Oct 03 2025 at 16:17):

I don't immediately see it either. One reduction I can see would involve discrete $\lambda$-graphs, parallel pairs of arrows, and the graph for $\lambda$-sized "wide pushouts". Another would involve discrete $\lambda$-graphs and the graph consisting of $\lambda$ parallel pairs of arrows all with the same codomain but different domains.

Elisha Goldman (Oct 03 2025 at 17:57):

Does the corresponding construction for (co)limits work?

Mike Shulman (Oct 03 2025 at 18:16):

I don't think so, because constructing the two maps between coproducts that you then coequalize requires the universal property of the domain coproduct, whereas the cocones that merely exist in a filtered category don't have a universal property.

Morgan Rogers (he/him) (Oct 04 2025 at 12:04):

You can take a cone under the objects in the diagram, coequalize all of the parallel arrows from each object to the nadir (producing a wide span), put a cone under the objects in that wide span, and finally coequalize all of the arrows from the original nadir to the new one.

Morgan Rogers (he/him) (Oct 04 2025 at 12:05):

Where "all of the arrows" refers respectively to the arrows in the diagrams composed with the cocone morphisms and then the arrows constructed from the coequalizing arrows and the cocone arrows.

Mike Shulman (Oct 04 2025 at 16:01):

Ah, that's what I was missing! Thanks Morgan. I recorded this at [[filtered category]].

Robin Jourde (Oct 06 2025 at 09:56):

Thank you all! And thanks Mike for writing it down on the nlab